# Z1 vs Dependency Ratio Correlations

## Purpose

Explains why Z1 (positive on savings) appears to contradict the negative effect of old_dep on savings: Z1 captures the full age distribution shape, including working-age bulge effects, not just old-age dependency.

## Cross-Correlations

| | Z_1 | Z_2 | Z_3 | old_dep | youth_dep | working_age_share | total_dep |
|---|---|---|---|---|---|---|---|
| Z_1 | 1.000 | 0.994 | 0.981 | 0.866 | -0.899 | 0.692 | -0.702 |
| Z_2 | 0.994 | 1.000 | 0.996 | 0.905 | -0.850 | 0.609 | -0.622 |
| Z_3 | 0.981 | 0.996 | 1.000 | 0.930 | -0.807 | 0.543 | -0.557 |
| old_dep | 0.866 | 0.905 | 0.930 | 1.000 | -0.653 | 0.312 | -0.331 |
| youth_dep | -0.899 | -0.850 | -0.807 | -0.653 | 1.000 | -0.917 | 0.931 |
| working_age_share | 0.692 | 0.609 | 0.543 | 0.312 | -0.917 | 1.000 | -0.992 |
| total_dep | -0.702 | -0.622 | -0.557 | -0.331 | 0.931 | -0.992 | 1.000 |

## Key Correlations

| Pair | Correlation | Interpretation |
|---|---|---|
| Z1, old_dep | 0.866 | Z1 captures aging but also working-age bulge |
| Z1, youth_dep | -0.899 | Z1 is strongly inversely related to youth dependency |
| Z1, working_age_share | 0.692 | Z1 tracks the lifecycle saving peak |

## Interpretation

Z1 (the first principal component of the age distribution) rises as countries transition from young to mature age structures. This captures *both* the decline in youth dependency (which raises savings) and the rise in old-age dependency (which lowers savings). Because the working-age bulge effect dominates at early stages of demographic transition, Z1 is positively associated with savings even though old_dep alone has a negative association.

The negative coefficient on old_dep in age-ratio regressions captures only the dissaving effect of aging. Z1's positive coefficient reflects the net lifecycle effect: as the entire age distribution shifts from young toward middle-aged, aggregate saving rises. Only at very advanced stages of aging (when old_dep dominates) does the dissaving channel prevail.
